In how many ways n distinct balls can be given to k children so that no child gets more than 3 balls?

What a Workbook!

Math example of each

Requirements:

1. Three different application problems involving setting up and solving a system of equations.
   1. Basic
   2. Mixture
   3. Distance, Rate, and Time
2. Three different application problems involving setting up and solving an equation using factoring of polynomials.
   1. Geometric
   2. Pythagorean Theorem
   3. Consecutive Integers
3. One problem per page.
   1. Clear Problem Statement
   2. Step by Step solution to problem
   3. Clear and detailed explanation of how solution was arrived at.
   4. Visual appeal

Use the procedures developed in this chapter to find the general solution of the differential equation.

x²y'' − 6xy' + 10y = −2x⁴ + 3x²

A computer manufacturer produces three types of computers: laptop, desktop, and tablet The manufacturer has designed their supply chain in such a way so that the final assembly of their computers entails putting together different quantities of the same parts. We will assume that the costs for parts unique to the model have already been accounted for in the reported per-unit profits, and that inventory levels for these other parts do not change the optimal allocation. We assume that there are three parts: CPU, RAM, and Hard Drive, each labels, respectively, as part 0, part 1, and part 2. The manufacturer would like to know how many of each model it should manufacture so as to maximize their total profit. However, if inventory of the raw materials is left over after sales within the planning period, then the manufacturer is charged a per-unit fee for each part. Use this information to construct a linear program and to answer the questions below

What is the set of constraints? (a) a0,0x0 + a0,1x1 + a0,2x2 + a0,3x3 + a0,4x4 ≤ b0 a1,0x0 + a1,1x1 + a1,2x2 + a1,3x3 + a1,4x4 ≤ b1 a2,0x0 + a2,1x1 + a2,2x2 + a2,3x3 + a2,4x4 ≤ b2 (b) a0,0x0 + a0,1x1 + a0,2x2 + a0,3x3 ≤ h0 a1,0x0 + a1,1x1 + a1,2x2 + a1,3x3 ≤ h1 a2,0x0 + a2,1x1 + a2,2x2 + a2,3x3 ≤ h2 a3,0x0 + a3,1x1 + a3,2x2 + a3,3x3 ≤ h3 a4,0x0 + a4,1x1 + a4,2x2 + a4,3x3 ≤ h4 (c) a0,0x0 + a0,1x1 + a0,2x2 ≤ b0 a1,0x0 + a1,1x1 + a1,2x2 ≤ b1 a2,0x0 + a2,1x1 + a2,2x2 ≤ b2 a3,0x0 + a3,1x1 + a3,2x2 ≤ b3 (d) a0,0x0 + a0,1x1 + a0,2x2 ≤ h0 a1,0x0 + a1,1x1 + a1,2x2 ≤ h1 a2,0x0 + a2,1x1 + a2,2x2 ≤ h2

2. Prove that |[0, 1]| = |(0, 1)|

Prove that Z_5 with addition and multiplication mod 5 is a field.

Q3. A)

I. Put the following sentences into their Clausal Elements:

Eg. John / danced / yesterday. S V A

i. Seeing is believing.

ii. I wish you good luck.

iii. What he said at the meeting was completely balderdash.

iv. Out of joy, the lecturer appointed Collins the course representative immediately.

v. Gyesi scores the third goal for Oasis FC in every competition.

II. Construct a Meaningful Sentence for EACH of the Following Patterns

i. SAVC

ii. ASVOC

iii. SVOA

iv. SVOCA

v. AASVC

5). Let f : [a,b] to R be bounded and f(x) > a > 0, for all x in [a,b]. Show that if f is Riemann integrable on [a,b] then 1/f : [a,b] to R, (1/f) (x) = 1/f(x) is also Riemann integrable on [a,b].

Solve Laplace’s equation w_xx + w_yy = 0 on the rectangle R = {(x, y) : 0 ≤ x ≤ a, 0 ≤ y ≤ b} subject to the boundary conditions w(x, 0) = 0, w(x, b) = 0, w(0, y) = f₁(y),  w(a, y) = f₂(y). Include coefficient formulas.

a spherical vessel with a diameter of 2cm when empty, has a mass of 2.00g. what is the greatest volume of water that can be placed in the vessel and still have the vessel float at the surface of water?

Create a hot and cold game in Matlab to find a point in an (X,Y) 2D coordinate plane. The user answers two inputs: x=Input('x') y=Input('y'). THE INPUTS WILL ONLY BE INTEGERS FROM 0 TO 100. This means no negative numbers. You write the program so the computer plays the game. The computer must play the game (comparing distances only) and give the same point as the user inputs. To keep it simple the code will check distances until distance = 0 (right on the point). certain section can be eliminated by comparing distances. You will use mostly loops and if-then statements. ONLY PROMPT THE USER TO INPUT THE "SECRET" X AND Y LOCATION once. THE COMPUTER WILL FIND THE POINT. DO NOT KEEP PROMPTING THE USER TO KEEP GUESSING.

A certain lake currently has an average trout population of 10,000. The population naturally oscillates above and below average by 1,000 every year. This year, the lake was opened to fishermen. If fishermen catch 2,000 fish every year, how long will it take for the lake to have no more trout? (Assume that the trout population at the beginning of the year is increasing. Round your answer to two decimal places.)

3.5.4 ([Ber14, Ex. 3.6.14]). Let T : V → W and S : W → U be linear maps, with V finite dimensional.

(a) If S is injective, then Ker ST = Ker T and rank(ST) = rank(T).

(b) If T is surjective, then Im ST = Im S and null(ST) − null(S) = dim V − dim W

Explain why we can have "even" and "odd" half range expansions.